Visionary mathematician Vladimir Voevodsky

A self-study Russian mathematician, kicked out 3 times in high schools, expelled from Moscow University, all because he did not attend classes, preferred to self-study in a broader scope for his curiosity, at his own faster speed than the rigid curriculum and boring test-and-exams regime in classrooms.

He did the PhD in Harvard by invitation even he did not have a Bachelor degree, and he barely passed the Harvard’s QE (Qualifying Exams) in Algebraic Geometry, a field in which he made a revolutionary discovery few years later, and for which he was awarded the highest honor : Fields Medal.

This is the typical trait of the geniuses like Evariste Galois, Albert Einstein, Ramanujian, Hua Luogeng (华罗庚), Zhang YiTang (张益唐 – proved “70m Twin Prime Gap”) [#] , Chen Jingrun (陈景润, proved Goldbach Conjecture “1+2”) [##]… with self-motivated curiosity in their field of passion, with reading from the Masters’ works by themselves, PLUS thinking on a problem over many years with perseverance, they finally made great discoveries !


  1. Topology 拓扑
  2. Algebraic Geometry 代数几何: Geometry and Algebraic Equations
  3. Motif: Grothendieck
  4. Homotopy Theory 同伦

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How (not) to memorise mathematics

Many excellent Math students after leaving universities more than 10 years forget 90% of math they learned, save some primary school arithmetics – few could do Singapore PSLE Modelling Math or solve quadratic equations.

The “Story-Telling” memory technique via “Signposts” can be used to reconstruct math from first principles:

Note: Lewis Carroll: the author of “Alice in wonderland”

Cambridge Professor Tim Gowers (Fields Medalist) suggested the similar pedagogy of “Memorise by First Principles”.

What is math? 

What is Math ? Interesting article below:

  • Mathematics = “that which is learned“ –(Pythagoras)

Math is not about calculation, it is understanding the nature, the universe, the philosophy (logic, intelligence – both “human” and “artificial”)…

What is Axiom, Lemma, Proposition ? Why rigorous Calculus was needed hundred years after Newton & Leibniz had invented it – “Epsilon-Delta” Analysis.

Difference between Riemann Integral & Lebesgue Integral ?

Unknown German Retiree Proved The “Gaussian Correlation Inequality” Conjecture

\boxed {P (a + b) \geq P (a) \times P (b)}

Case “=” : if (a, b) independent 
Case “>” : if (a, b) dependent

Thomas Royen used only high-school math (function, derivative) in his proof in 2014. He then published it in website – like Perelman did with the “Poincaré Conjecture”.

The genius who rejected Fields Medal & Clay Prize: Grigori Perelman 

Russian Jewish mathematician Grigori Perelman proved the Poincaré Conjecture in 7 years of solitude research in his Russian apartment  – same 7 years of solitude for Andrew Wiles (The Fermat’s Last Theorem) in the Cambridge attic house and Zhang Yitang  张益唐 (7-Million-Gap Twin Primes) in the “Subway” sandwich kitchen.

7 is the Perfect Number. 1 week has 7 days,  according to the “Book of Genesis”, God created the universe in 6 days and rested “Sabbath” on the beautiful 7th day.

People involved in his journey: 

  • His mother who supported his early education in Math, all the way to his international fame and adulthood — Erdös Paul also had a mother supporting her single son in the entire Mathematical life.
  • Prof Hamilton who “unintentionally” gave him the inspiration of the “Ricci-flow” tool – the first key to the door of The Poincaré Conjecture.
  • His closed Chinese friend the MIT Math Prof  田刚 (Tian Gang) – a PhD student of Prof S.T. Yau.
  • Fields Medal which he rejected — reason being he felt Prof Hamilton deserved a share of the credit.
  • Clay Prize which he rejected the U$ 1 million — for the same reason as the Fields Medal.
  • Prof  S.T. Yau (邱成桐) ‘s lawsuit against “The New Yorker” jounalists on “The Poincaré Conjecture” article regarding the purported claims by his 2 Chinese mathematician students.